Complex potential formulation of the axially symmetric gravitational field problem

Abstract

Spin-coefficients and null tetrad components of the Ricci tensor and the Weyl conform tensor are evaluated in terms of a single complex gravitational potential ε, while null tetrad components of the electromagnetic stress energy tensor are evaluated in terms of a second complex potential φ. All the results are expressed elegantly in terms of a differential operator ð, similar to the ``thop'' of Newman and Penrose. The problem of finding physically pertinent stationary axially symmetric Einstein-Maxwell fields is reduced to the search for a complex solution ξ0(x, y) of one nonlinear differential equation subject to simple subsidiary conditions.